146 
MR. W. F. SHEPPARD OH THE APPLICATIOH OF THE THEORY OF 
For practical applications of this method, it is sufficient to have the single figure 
as shown in tig. 7. The curves representing the displacement of the figure through 
the distance D/^tt can then be traced by means of a double-barred parallel rule 
or a,n antigraph. But it is better to draw the curves directly from Tables III. 
and IV.* 
§ 29. Differential Relation of V and D.—Let Y denote the proportion of individuals 
for which L exceeds X, and M exceeds Y. Then V is the volume of the solid 
lying on the positive side of the vertical planes drawn through N and n (§ 28) at 
right angles to ON and On respectively. Let the sections of the solid by these 
planes intersect in the ordinate WR ; and let them meet the base-plane in the 
straight lines NW 77 and nW^ respectively. 
Let V — V denote the value which V would have if the divero’ence, instead of beino- 
D, were D 6, the values of ON and On being unaltered. This alteration in the 
value of V might be obtained by keeping ON and Nt 7 fixed, and rotating On and 
n^ about OZ through an angle 9. Now suppose that 6 is very small. Then the 
consecutive positions of the vertical section through will intersect close to the 
ordinate at n ; and therefore v is the volume obtained by rotating the area 
about the ordinate at n through an angle 9. Hence, for a first approximation, we 
have V = WR.0 (§ 5). 
We might have obtained this result by considering the alteration, due to the 
change of I) into D + d, of the diagram constructed in the manner explained in the 
last section. The area which is equal to V is bounded by the base and by two curves 
intersecting at a point whose height above the base is 27r.WR ; and the decrement 
V is obtained by shifting one curve laterally through a distance 0/27r. Hence 
V = WR.0. Let the two curves, at their point of intersection, be inclined to the 
base at angles and w,- Then it will be seen that for a second approximation we 
have V = WR. 9 ^ sin wj sin m., cosec (wi + 0 ) 2 }. f/'liry. 
The ordinate WR is the ordinate, for abscissa (x“ — 2 a;y cosD + y“)- cosec D, of 
* It lias beeu suggested that tlie one set of curves miglit be drawn on a board or stiff card, and the 
other on a thin sheet of some transparent substance (e.gf.,of talc), which could be slipped across the face 
of the card. Tliis, however, might require the curves to be drawn on too small a scale to be really 
useful. 
Table HI. can be used for drawing the curve corresponding to any value of a not given in the table. 
If X and X are the abscisste of the standard curve corresponding to class-indices a and a', the equations to 
the corresponding curves of the divergence-diagram are 0 = exp( — AHsec’27r0) and z'= exp( —W’sec^ 2 -d). 
Hence, for any particular value of 0, we have log z'jlogz = The value of z being given by the 
table, the value of z' may be deduced by means of an ordinary slide-rule and a pair of proportional 
compasses. 
The methods described in the text can be extended to the problems which occur in the theory of the 
error in the position of a point in a plane (as in Beavais’ memoir, referred to bj' Professor Peaesox). 
Thus the condition that the j^oint lies within an area limited by the curve / (x, y) = 0 is found b}' 
taking the curve 2 of § II to be the curve whose equation, referred to axes including an angle tt — D, 
is/ (r/usin D, y/bsin D) = 0, and then counting the dots or measuring the corresponding cylinder-area. 
